Contextualization

The world around us is full of phenomena that can be mathematically described. In nature, technology, finance, and even in the population growth of a city, we can find situations whose description involves the concept of exponential function. Therefore, understanding and mastering this concept facilitates the interpretation and prediction of real-world phenomena, and that is precisely what we will explore in this project.

The Exponential Function is one of the fundamental concepts of mathematics and has a wide range of applications, such as calculating compound interest in the field of financial mathematics, studying population growth in biology, predicting the evolution of an epidemic in public health, and even in physics, through the law of radioactive decay.

Introduction

An exponential function is one in which the variable appears in the exponent of a constant, which we call the base. This is a powerful concept in mathematics because it paves the way to represent growth and decay that are proportional to the current value of the variable - that is, the higher the value, the greater the growth or decay.

This behavior is fundamental for understanding various natural and social phenomena. For example, compound interest is calculated based on an exponential function, as well as the population growth of a species without environmental restrictions.

Graphically, the exponential function forms a curve that grows (or decreases, depending on the base) rapidly. Understanding this graph, and the relationship between the curve and the function itself, is essential to use the concept effectively.

Practical Activity

Activity Title: The Power of Exponential

Project Objective

Investigate real cases of exponential growth and decay, understand the behavior of the graphs of these functions and their fundamental characteristics, and finally, present the understanding of the topic in a creative and collaborative way.

Detailed Project Description

This activity aims to illustrate the application of exponential functions in the real world. Students will work in groups of 3 to 5 and will play the role of mathematics researchers, investigating and presenting the concepts and applications of exponential functions and their graphs.

The work will comprise two main areas:

1. Theoretical study: The group should delve into the theory of exponential functions, focusing on the characteristics of the curve generated by these functions (growth, decay, domain, image).

2. Real cases: Using theoretical knowledge, students should identify, research, and explain a real case where exponential growth or decay is observed. It can be something in biology (population growth), physics (radioactive decay), or economics (compound interest).

Required Materials

• Books and online resources for research.
• Graph creation software (it can be Excel spreadsheet, GeoGebra, specific mathematics software, or even free apps available on the internet).
• Slide presentation for the exposition of the studied case.

Detailed Step-by-Step

1. Theoretical study: Students should research in books and on the internet about exponential functions, especially on how these functions are graphically represented. It is important that students understand the concept and characteristics of exponential functions well.

2. Identification of real cases: Next, the group must identify at least one real case where exponential functions apply. It can be cases of population growth, radioactive decay, compound interest, etc.

3. Research and case development: After choosing the case, the group should research in-depth on the topic. How do exponential functions apply in this case? What would be the graph of this function? How does the function explain the observed behavior in the real case?

4. Report elaboration: The group should then write a complete report, following the guidelines: Introduction, Development, Conclusions, and Bibliography used.

5. Presentation preparation: Upon completing the report, the group should prepare a slide presentation that will be presented to the class. The presentation should synthesize the work done, highlighting the main points of the research and the conclusions reached.

6. Presentation: Students will present the project to the class, explaining the concept of exponential function, the studied real case, and the conclusions reached.

Project Deliverables

The groups must deliver:

1. Written report: It must contain all the requested topics, with a detailed description of the studied real case, the applied exponential function, the analysis of the graph, and the relationship between the function and the studied case. In addition, students must detail the work process, the resources used, the difficulties faced, as well as the solutions found.

2. Slide presentation: The presentation should be clear and concise, highlighting the main points of the research and the conclusions. The group must be prepared to answer questions and explain the presented content. The presentation should have a maximum duration of 15 minutes.

Throughout the work, it is essential that students continue to reinforce socio-emotional skills, such as effective communication, collaboration, time management, and problem-solving.